> [!definition] > > Let $E$ be a [[Vector Space|vector space]] over $K \in \bracs{\real, \complex}$ and $\seqi{E}$ with $\bracsn{T_j^i: i, j \in I, i \lesssim j}$ be a [[Directed Set|directed system]] of [[Topological Vector Space|topological vector spaces]], then there exists a topological vector space $E$ and a family of continuous linear maps $\bracsn{T^i_E: i \in I}$ such that > 1. $T^i_E: E_i \to E$ is a continuous linear map. > 2. $T^i_j \circ T^j_E = T^i_E$ for all $i, j \in I$. > 3. If $F$ is another topological vector space, and $T: E \to F$ is a linear map, then $T$ is continuous if and only if $T \circ T^i_E$ is continuous for each $i \in I$. > 4. If $(F, \bracsn{T_F^i: i \in I})$ is another pair satisfying $(1)$ and $(2)$, then there exists a unique continuous linear map $T^E_F: E \to F$ such that $T^i_F = T^E_F \circ T^i_E$ for all $i \in I$. > > In other words, [[Direct Limit|direct limits]] exist in the category of topological vector spaces over $K$. > > *Proof*. $(1)$, $(2)$, $(4)$, existence: Let $(E, \bracsn{T^i_E})$ be the direct limit of $(\seqi{E}, \bracsn{T^i_j})$ in the category of $K$-vector spaces, then $E$ satisfies the universal property in the category of $K$-vector spaces, and the factor maps exist. It's sufficient to induce a topology on $E$ on which each $T^i_E: E_i \to E$ and $T^E_F$ in $(3)$ are continuous. > > $(1)$, continuity: Let $\mathscr{T} = \seqi{\mathcal T}$ be the collection of all topologies on $E$ such that $E$ is a topological vector space and $T^i_E: E_i \to E$ is continuous for each $i \in I$. Note that $\mathscr{T}$ is non-empty as the trivial topology satisfies the given criteria. Let $\mathcal T$ be the [[Initial Topology on Topological Vector Space|initial topology]] generated by the identity $I: E \to (E, \mathcal T_i)$ into each topology, then $\mathcal T$ is a topology on $E$ with $T^i_E: E_i \to (E, \mathcal T)$ being continuous for all $i \in I$. > > $(3)$: Let $F$ be a topological vector space and $T: E \to F$ be a linear map such that $T \circ T^i_E$ is continuous for each $i \in I$, then the topology on $E$ generated by $F$ belongs to $\mathscr{T}$. Therefore $T$ is continuous. > > $(4)$: Follows directly from $(3)$.